Sparsity in Multivariate Extremes with Applications to Anomaly Detection
This work addresses a major bottleneck in risk management fields like finance and environmental science by providing a scalable approach for analyzing rare events, though it is incremental in applying sparsity to an existing framework.
The paper tackles the challenge of modeling the angular measure in high-dimensional multivariate extreme value theory by proposing a novel methodology that exhibits sparsity patterns in the dependence structure of extremes, enabling scaling up of existing methods and leading to a first anomaly detection algorithm based on multivariate EVT with strong empirical evidence.
Capturing the dependence structure of multivariate extreme events is a major concern in many fields involving the management of risks stemming from multiple sources, e.g. portfolio monitoring, insurance, environmental risk management and anomaly detection. One convenient (non-parametric) characterization of extremal dependence in the framework of multivariate Extreme Value Theory (EVT) is the angular measure, which provides direct information about the probable 'directions' of extremes, that is, the relative contribution of each feature/coordinate of the 'largest' observations. Modeling the angular measure in high dimensional problems is a major challenge for the multivariate analysis of rare events. The present paper proposes a novel methodology aiming at exhibiting a sparsity pattern within the dependence structure of extremes. This is done by estimating the amount of mass spread by the angular measure on representative sets of directions, corresponding to specific sub-cones of $R^d\_+$. This dimension reduction technique paves the way towards scaling up existing multivariate EVT methods. Beyond a non-asymptotic study providing a theoretical validity framework for our method, we propose as a direct application a --first-- anomaly detection algorithm based on multivariate EVT. This algorithm builds a sparse 'normal profile' of extreme behaviours, to be confronted with new (possibly abnormal) extreme observations. Illustrative experimental results provide strong empirical evidence of the relevance of our approach.